Engineers and scientists are increasingly turning to sophisticated computer-based simulation models to predict and optimize process behavior in virtual environments. Yet, to be effective, simulation models need to have a high degree of accuracy to be reliable. An important part of simulation development, therefore, is parameter adaptation wherein the values of model parameters (i.e., model coefficients and exponents) are adjusted so as to maximize the accuracy of simulation relative to the experimental data available from the process. If parameter adaptation leads to acceptable predictions, the model is declared valid. Otherwise, the model is refined to higher levels of complexity so as to provide an expanded basis for parameter adaptation. The availability of an efficient and reliable parameter adaptation method, therefore, is crucial to model validation and simulation development.
Methods of parameter adaptation typically seek solutions to a set of parameters e that minimize a loss function V(Θ) over the sampled time T, as set forth in the relation:{circumflex over (Θ)}={circumflex over (Θ)}(ZN=argΘmin V(Θ,ZN)  (1)where ZN comprises the measured outputs y(t), and
      V    ⁡          (              Θ        ,                  Z          N                    )        =            ∫      Q      T        ⁢                  L        ⁡                  (                      ∈                          (              t              )                                )                    ⁢                          ⁢              ⅆ        t            is a scalar-valued (typically positive) function of the prediction error E(t)=y(t)−y(t) between the measured outputs y(t) and model outputs y(t)=Mθ(u(t)) with u(t) being the input applied to the process. In cases where the process can be accurately represented by a model that is linear in parameters, or when the nonlinear model is transformed into a functional series that is linear in parameters, each model output can be defined as a linear function of the parameters to yield an explicit gradient of the loss function in terms of the parameters for regression analysis. Otherwise, parameter adaptation becomes analogous to a multi-objective (due to multiplicity of outputs) nonlinear optimization, wherein the solution is sought through a variety of methods, such as gradient-descent, genetic algorithms, convex programming, Monte Carlo, etc. In all these error minimization approaches a search of the entire parameter space is conducted for the solution.
However, not all model parameters are erroneous. Neither is the indiscriminate search of the entire parameter space practical for complex simulation models. As a remedy, experts use a manual approach of selecting parameters that they speculate to most effectively reduce each prediction error. They usually select the suspect parameters by the similarity of the shape of their dynamic effect to the shape of the transient prediction error. They then alter these parameters within their range in the direction that are expected to reduce the error and run new simulations to evaluate the effect of these parameter changes. If the new parameter values improve the results, they are further adjusted until they no longer improve the simulation. This process is followed for another set of suspect parameters and repeated for all the others until satisfactory results are obtained. If at the end of parameter adaptation adequate precision is attained, the model is declared valid and the adjusted model is presumed to represent the process. Even though this manual approach lacks a well-defined procedure and is tedious and time-consuming, it provides the advantage of targeted (selective) adaptation wherein each parameter is adapted separately.
A continuing need exists, however, for further improvements in model development and operation.